SPECTRAL NORMS OF CIRCULANT-TYPE MATRICES WITH BINOMIAL COEFFICIENTS AND HARMONIC NUMBERS

2014 ◽  
Vol 11 (05) ◽  
pp. 1350076 ◽  
Author(s):  
JIANWEI ZHOU ◽  
ZHAOLIN JIANG
Keyword(s):  
Circulant Matrices ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Numerical Tests ◽  
Skew Circulant

In this paper, we investigate spectral norms for circulant-type matrices, including circulant, skew-circulant, and g-circulant matrices. The entries are product of Binomial coefficients with Harmonic numbers. We obtain explicit identities for these spectral norms. Employing these approaches, we list some numerical tests to verify our results.

scholarly journals The Explicit Identities for Spectral Norms of Circulant-Type Matrices Involving Binomial Coefficients and Harmonic Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Jianwei Zhou ◽  
Xiangyong Chen ◽  
Zhaolin Jiang
Keyword(s):  
Circulant Matrix ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Numerical Tests ◽  
Skew Circulant ◽  
Explicit Formulae

The explicit formulae of spectral norms for circulant-type matrices are investigated; the matrices are circulant matrix, skew-circulant matrix, andg-circulant matrix, respectively. The entries are products of binomial coefficients with harmonic numbers. Explicit identities for these spectral norms are obtained. Employing these approaches, some numerical tests are listed to verify the results.

scholarly journals The Identical Estimates of Spectral Norms for Circulant Matrices with Binomial Coefficients Combined with Fibonacci Numbers and Lucas Numbers Entries

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Jianwei Zhou
Keyword(s):  
Fibonacci Numbers ◽  
Circulant Matrices ◽  
Binomial Coefficients ◽  
Lucas Numbers ◽  
Numerical Tests

Improved estimates for spectral norms of circulant matrices are investigated, and the entries are binomial coefficients combined with either Fibonacci numbers or Lucas numbers. Employing the properties of given circulant matrices, this paper improves the inequalities for their spectral norms, and gets corresponding identities of spectral norms. Moreover, by some well-known identities, the explicit identities for spectral norms are obtained. Some numerical tests are listed to verify the results.

scholarly journals Harmonic Numbers and Cubed Binomial Coefficients

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Anthony Sofo
Keyword(s):  
Closed Form ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Polygamma Functions ◽  
Form Representation ◽  
The Given

Euler related results on the sum of the ratio of harmonic numbers and cubed binomial coefficients are investigated in this paper. Integral and closed-form representation of sums are developed in terms of zeta and polygamma functions. The given representations are new.

scholarly journals The Determinants, Inverses, Norm, and Spread of Skew Circulant Type Matrices Involving Any Continuous Lucas Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Jin-jiang Yao ◽  
Zhao-lin Jiang
Keyword(s):  
Circulant Matrix ◽  
Circulant Matrices ◽  
Lucas Numbers ◽  
Transformation Matrices ◽  
Skew Circulant ◽  
The Relationship

We consider the skew circulant and skew left circulant matrices with any continuous Lucas numbers. Firstly, we discuss the invertibility of the skew circulant matrices and present the determinant and the inverse matrices by constructing the transformation matrices. Furthermore, the invertibility of the skew left circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the skew left circulant matrices by utilizing the relationship between skew left circulant matrices and skew circulant matrix, respectively. Finally, the four kinds of norms and bounds for the spread of these matrices are given, respectively.

scholarly journals Four-Dimensional Almost Einstein Manifolds with Skew-Circulant Structres

2019 ◽  
Author(s):  
Iva Dokuzova ◽  
Dimitar Razpopov
Keyword(s):  
Tangent Space ◽  
Local Coordinate System ◽  
Complex Structure ◽  
Hermitian Manifold ◽  
Circulant Matrices ◽  
Fourth Power ◽  
Einstein Manifolds ◽  
Almost Hermitian Manifold ◽  
Riemannian Connection ◽  
Skew Circulant

We consider a four-dimensional Riemannian manifold M equipped with an additional tensor structure S, whose fourth power is minus identity and the second power is an almost complex structure. In a local coordinate system the components of the metric g and the structure S form skew-circulant matrices. Both structures S and g are compatible, such that an isometry is induced in every tangent space of M. By a special identity for the curvature tensor, generated by the Riemannian connection of g, we determine classes of an Einstein manifolds and an almost Einstein manifolds. For such manifolds we obtain propositions for the sectional curvatures of some special 2-planes in a tangent space of M. We consider an almost Hermitian manifold associated with the studied manifold and find conditions for g, under which it is a Kähler manifold. We construct some examples of the considered manifolds on Lie groups.

scholarly journals Congruences involving alternating sums related to harmonic numbers and binomial coefficients

2020 ◽  
Vol 26 (4) ◽  
pp. 39-51
Author(s):  
Laid Elkhiri ◽  
◽  
Miloud Mihoubi ◽  
Abdellah Derbal ◽  
◽  
...  
Keyword(s):  
Prime Number ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Arithmetic Properties ◽  
Alternating Sums ◽  
Generalized Harmonic Numbers

In 2017, Bing He investigated arithmetic properties to obtain various basic congruences modulo a prime for several alternating sums involving harmonic numbers and binomial coefficients. In this paper we study how we can obtain more congruences modulo a power of a prime number p (super congruences) in the ring of p-integer \mathbb{Z}_{p} involving binomial coefficients and generalized harmonic numbers.

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